Julio Rodrigo Fernández

The main goal of this article is to study, from a game theory perspective, the composition of the Spanish Parliament according to Article 68 of the Spanish Constitution, although the proposed model is applicable to the reduction of other representation chambers. It even allows for the sporadic design of the chamber in periods of crisis such as those we are currently experiencing. We use power indices to analyse feasible allocations of seats among the circumscriptions, modifying the size of the Parliament and considering different minimum initial numbers of seats per province. We propose two modifications of the composition following the cubic root rule of the de jure population. Finally, we compare the results of the general elections of December 2015 and June 2016 for the election of the members of the Congress of Deputies (Spanish Parliament) with the current system, which distributes a total number of 350 deputies among the provinces, with an initial minimum of two deputies (system 350/2), with another distribution system that distributed 360 deputies with an initial minimum of one deputy per province (system 360/1).

A cooperative game represents a situation in which a set of agents form coalitions in order to achieve a common good. To allocate the benefits of the result of this cooperation there exist several values such as the Shapley value or the Banzhaf value. Sometimes it is considered that not all communications between players are feasible and a graph is introduced to represent them. Myerson (1977) introduced a Shapley-type value for these situations. Another model for cooperative games is the Owen model, Owen (1977), in which players that have similar interests form a priori unions that bargain as a block in order to get a fair payoff. The model of cooperation introduced in this paper combines these two models following Casajus (2007). The situation consists of a communication graph where a two-step value is defined. In the first step a negotiation among the connected components is made and in the second one players inside each connected component bargain. This model can be extended to fuzzy contexts such as proximity relations that consider leveled closeness between agents as we proposed in 2016. There are two extensions of the Banzhaf value to the Owen model, because the natural way loses the group symmetry property. In this paper we construct an appropriate value to extend the symmetric option for situations with a proximity relation and provide it with an axiomatization. Then we apply this value to a political situation.

A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. In order to obtain this allocation, a characteristic function is known. It establishes the profit of each coalition of agents if this coalition decides to act alone. Originally players are considered symmetric and then the allocation only depends on the characteristic function; this paper is about cooperative games with an asymmetric set of agents. We introduced cooperative games with a soft set of agents which explains those parameters determining the asymmetry among them in the cooperation. Now the characteristic function is defined not over the coalitions but over the soft coalitions, namely the profit depends not only on the formed coalition but also on the attributes considered for the players in the coalition. The best known of the allocation rules for cooperative games is the Shapley value. We propose a Shapley kind solution for soft games.